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As everyone else has said, it means "just over x amount" in this context. The major hint for "just over" is 才, meaning "merely", or "just". So, they could be saying "I've only been here just over three months... give me a break!"

Another thing you may see to indicate "it's just over", or "it's been about this long" is the use of 左右 zuo3 you4 ("left, right").

In this case, "多" can be taken to mean "plus". Plus, as an adjective, carries the meaning of "more than, by a certain amount".

I have only been here for three months plus.

I have to disagree with what many have written here regarding the purpose of "才". Using the following example as a form of disambiguation:

我三点多有一个会议，现在才十一点多，我们还有时间吃午饭。
I have a meeting at three plus, now is only 11 plus, we still have
time for lunch.

"才" has no implication on the actual time since "多" in 三点多 is not much different from "多" in 十一点多. In the context of your sentence, "才" is merely used to emphasize the shortness of the time period that you have been here. In my example, "才" is used to emphasize the earliness of time.

"多" by itself means exceeding by less than a unit of measurement. Example:

Put it in a simple way, first, we know 多 means many. But the 多 here has a meaning of many or less - 多少。 So literally is 3 months and " many or less". Because the speaker is not sure many or less, so here 三个多月 in English term is 3 months or so.